Methods and apparatuses for photovoltaic power management

ABSTRACT

Methods and apparatuses for equalizing voltages across a plurality of photovoltaic units connected in series are provided. The apparatus may include a plurality of energy storage devices. In a first configuration, each of the energy storage devices is configured to be connected in parallel with one of a first set of the photovoltaic units. In a second configuration, each of the energy storage devices is configured to be connected in parallel with one of a second set of the photovoltaic units. The apparatus may also include a plurality of switches configured to switch between the first configuration and the second configuration, to equalize the voltages across the photovoltaic units.

CROSS-REFERENCE TO RELATED APPLICATIONS

This application claims priority under 35 U.S.C. §119 to U.S. Provisional Patent Application No. 61/216,101, filed on May 13, 2009, the contents of which are hereby incorporated by reference in their entirety.

BACKGROUND OF THE INVENTION

The present invention relates to methods and apparatuses for power management in photovoltaic (PV) installations. FIG. 1 is a high-level circuit diagram of a typical related-art PV installation 10. As shown in FIG. 1, the PV installation 10 includes a plurality of PV modules 20 connected in series and in parallel. The PV modules 20 in each column are connected in series, and the columns of PV modules 20 are connected in parallel to form the PV installation 10. For example, a column may include approximately 20 PV modules 20, and the PV installation 10 may include approximately 250 columns. As shown in FIG. 1, typical PV installations use a single centralized inverter 30 to convert the DC output of the PV installation 10 to AC power to be provided to the grid 40. A large-scale grid-connected PV installation 10 may generate anywhere between 500 W and 5 kW for residential systems to over 1 MW for PV power plant applications.

For PV power plant applications, the input DC voltage of the centralized inverter 30 is typically around 500 V, and the output AC voltage may be on the order of 120 V up to 100 kV. The centralized inverter 30 shapes the AC output voltage or current to obtain a clean 60 Hz sinusoidal waveform with strictly regulated noise and distortion limits. The centralized inverter 30 also manages the DC voltage across the PV installation 10. There is typically an optimum DC voltage for the PV installation 10 known as the maximum power voltage (MPV). This is the DC voltage that coincides with the maximum power point (MPP) of the PV installation 10, which is the operating point where the PV installation 10 produces the maximum power. The MPV also coincides with a maximum power current (MPC), which is the current that coincides with the MPP. The centralized inverter 30 can typically regulate the power flowing out of the PV installation 10 to drive the PV installation 10 to the MPP with a maximum power point tracking (MPPT) algorithm.

With a single centralized inverter 30 managing the entire PV installation 10, all of the columns of PV modules 20 within the PV installation 10 operate with the same voltage, because they are connected in parallel. This can be a problem if the MPVs of the PV modules 20 are different. Additionally, within the columns, each of the PV modules 20 operates with the same series current, because they are connected in series. This can be a problem if the MPCs of the PV modules 20 are different. Although each PV module 20 is designed to have the same characteristics, the MPP, MPV, and MPC of a PV module 20 can change if the PV module 20 encounters local shading, dirt, or debris. Further, there can be a factory mismatch between two PV modules 20, in which the characteristics of the two PV modules 20 are slightly different as manufactured.

The PV module 20 in the lower left-hand corner of the PV installation 10 is shown in FIG. 1 as being partially shaded. This shading causes shifts in the MPV and MPC of the shaded PV module 20 such that the PV module 20 may be forced to a sub-optimal operating point. In some cases, the PV module 20 may contribute nearly zero power, even with only small amounts of partial shading. For example, if only 20% of the incident light is blocked from reaching the PV module 20, the characteristics of the shaded PV module can change substantially relative to the un-shaded PV modules 20.

Because the PV modules 20 are connected in series, they must have the same operating current. If the shaded PV module 20 is forced to the MPC of the un-shaded PV modules but the shaded PV module 20 does not have sufficient current available to support that MPC, then the voltage of the shaded PV module 20 will collapse and the shaded PV module 20 will contribute zero power. Alternatively, if the unshaded PV modules 20 are forced to the MPC of the shaded PV module 20, the unshaded PV modules 20 will operate at a current below their optimal MPC, and contribute less than the maximum power available. In this case, many of the unshaded PV modules 20 may operate sub-optimally because of the shading of one PV module 20. In both cases, the PV installation 10 operates sub-optimally because the centralized inverter 30 regulates many PV modules in aggregate, although the individual PV modules 20 may have varied characteristics. Accordingly, an apparatus and method are needed to address the problems caused by shading of PV modules 20 in the PV installation 10.

SUMMARY OF THE INVENTION

Apparatuses and methods consistent with the present invention equalize voltages across a plurality of photovoltaic units connected in series. According to an aspect of the invention, there is provided an apparatus that includes a plurality of energy storage devices. In a first configuration, each of the energy storage devices is configured to be connected in parallel with one of a first set of the photovoltaic units. In a second configuration, each of the energy storage devices is configured to be connected in parallel with one of a second set of the photovoltaic units. The apparatus also includes a plurality of switches configured to switch between the first configuration and the second configuration, to equalize the voltages across the photovoltaic units.

Each of the energy storage devices may have an impedance that is resonant at one or more frequencies. Each of the energy storage devices may include an inductor and a capacitor that are connected in series. The inductor may include a discrete inductor on a printed circuit board, interconnect layers of a printed circuit board, interconnect layers of an integrated circuit, bondwires that connect an integrated circuit and a printed circuit board, and/or interconnect wiring that connects printed circuit boards. Each of the energy storage devices may be a capacitor. Each of the energy storage devices may include a capacitor that has a substantial effective series inductance.

The apparatus may also include a controller configured to output a drive signal to the switches at a switching frequency to control switching of the switches. The switching frequency or a harmonic of the switching frequency may equal a resonant frequency of at least one of the energy storage devices.

Each of the photovoltaic units may include a photovoltaic cell. Alternatively, each of the photovoltaic units may include a string of photovoltaic cells that are connected in series. Alternatively, each of the photovoltaic units may include a module that includes at least one string of photovoltaic cells that are connected in series.

The apparatus may also include a plurality of bypass energy storage devices, each of which is configured in parallel with one of the photovoltaic units. The bypass energy storage devices may be capacitors. Each of the switches may include a single-pole, double-throw switch that includes at least one MOSFET.

The apparatus may also include a power point tracking converter that identifies an aggregate maximum power point for the plurality of photovoltaic units. The voltages across the photovoltaic units may be determined based on the aggregate maximum power point. The power point tracking converter may also identify an individual maximum power point for each of the plurality of photovoltaic units, in which case the voltages across the photovoltaic units are determined based on the aggregate maximum power point and the individual maximum power points. The power point tracking converter may identify the aggregate maximum power point for the plurality of photovoltaic units and another set of photovoltaic units that is connected in parallel with the plurality of photovoltaic units.

The apparatus may also include a connecting circuit configured to provide a connection to an external apparatus. The connecting circuit may include a connecting energy storage device and a connecting switch connected to the connecting energy storage device. In the first configuration, the connecting energy storage device is configured to be connected in parallel with a photovoltaic unit of the external apparatus via the connecting switch. In the second configuration, the connecting energy storage device is configured to be connected in parallel with one of the photovoltaic units of the apparatus via the connecting switch. The connecting circuit may also include a connecting bypass energy storage device, which is configured to filter the voltage across the photovoltaic unit of the apparatus that is connected in parallel with the connecting energy storage device in the second configuration.

According to another aspect of the invention, there is provided a multi-stage apparatus for equalizing voltages across a plurality of photovoltaic modules connected in series. Each of the photovoltaic modules includes a plurality of photovoltaic units connected in series. The multi-stage apparatus includes a plurality of apparatuses, each of which includes a plurality of energy storage devices. In a first configuration, each of the energy storage devices is configured to be connected in parallel with one of a first set of the photovoltaic units within one of the modules. In a second configuration, each of the energy storage devices is configured to be connected in parallel with one of a second set of the photovoltaic units within the one of the modules. The multi-stage apparatus also includes a plurality of switches configured to switch between the first configuration and the second configuration, and a connecting circuit configured to provide a connection between the apparatus and an adjacent apparatus among the plurality of apparatuses. The connecting circuit includes a connecting energy storage device and a connecting switch connected to the connecting energy storage device. In the first configuration, the connecting energy storage device is configured to be connected in parallel with a photovoltaic unit of the adjacent apparatus via the connecting switch. In the second configuration, the connecting energy storage device is configured to be connected in parallel with one of the photovoltaic units of the apparatus via the connecting switch. The connecting circuit also includes a plurality of connectors configured to connect each of the apparatuses to an adjacent one of the apparatuses. Switching between the first configuration and the second configuration equalizes the voltages across the photovoltaic units of all of the photovoltaic modules.

According to another aspect of the invention, there is provided a method of equalizing voltages across photovoltaic units connected in series. In a first configuration, the method includes storing a first energy from the photovoltaic units in a plurality of energy storage devices, wherein each of the energy storage devices is connected in parallel with one of a first set of the photovoltaic units. In a second configuration, the method includes storing a second energy from the photovoltaic units in the plurality of energy storage devices, wherein each of the energy storage devices is connected in parallel with one of a second set of the photovoltaic units. The method also includes switching between the first configuration and the second configuration until the voltages across the photovoltaic units are substantially equal. A frequency of the switching between the first configuration and the second configuration may equal a resonant frequency of the energy storage devices.

After the switching between the first configuration and the second configuration, the method may also include sending an energy equal to a difference between the first energy and the second energy to a plurality of bypass energy storage devices.

The method may also include identifying an aggregate maximum power point for the plurality of photovoltaic units, and determining the voltages across the photovoltaic units based on the aggregate maximum power point. The method may also include identifying an individual maximum power point for each of the plurality of photovoltaic units, in which case the voltages across the photovoltaic units are determined based on the aggregate maximum power point and the individual maximum power points. The aggregate maximum power point may be identified for the plurality of photovoltaic units and another set of photovoltaic units that is connected in parallel with the plurality of photovoltaic units.

According to another aspect of the invention, an apparatus for equalizing voltages across a plurality of photovoltaic units connected in series is provided. The apparatus includes a plurality of energy storage devices. In a first configuration, a first energy storage device is configured to be connected in parallel with a first photovoltaic unit, and a second energy storage device is configured to be connected in parallel with a second photovoltaic unit. In a second configuration, the first energy storage device is configured to be connected in parallel with the second photovoltaic unit, and the second energy storage device is configured to be connected in parallel with a third photovoltaic unit. A plurality of switches configured to switch between the first configuration and the second configuration, to equalize the voltages across the photovoltaic units.

According to another aspect of the invention, an apparatus for equalizing voltages across a plurality of photovoltaic units connected in series is provided. The apparatus includes an energy storage device. In a first configuration, the energy storage device is configured to be connected in parallel with a first one of the photovoltaic units. In a second configuration, the energy storage device is configured to be connected in parallel with a second one of the photovoltaic units. A switch is configured to switch between the first configuration and the second configuration, to equalize the voltages across the photovoltaic units.

Other objects, advantages, and novel features of the present invention will become apparent from the following detailed description of the invention when considered in conjunction with the accompanying drawings.

BRIEF DESCRIPTION OF THE DRAWINGS

FIG. 1 is a high-level circuit diagram of a typical related art PV installation with series and parallel strings of PV modules connected to a centralized inverter;

FIG. 2A shows a lumped circuit equivalent of a typical related art PV module with strings of series-connected PV cells and bypass diodes to short out strings under local shading conditions;

FIG. 2B is a Current-Voltage-Power curve of a PV module;

FIG. 2C shows the performance of two mismatched PV modules connected in series;

FIG. 2D shows the performance of two mismatched PV modules connected in parallel;

FIG. 3 shows a top-level schematic representation of a converter that includes a voltage equalization circuit adapted to be used in conjunction with a PV module in accordance with an exemplary embodiment of the invention;

FIG. 4 shows a conceptual overview of the operation of the voltage equalization circuit shown in FIG. 3;

FIG. 5 shows an extension of the voltage equalization circuit shown in FIG. 4 to a PV module with a number N of series-connected strings of PV cells;

FIG. 6A shows simulation results for individual string currents as functions of the total series current through the entire PV module shown in FIG. 4 while the voltage equalization circuit is operating and String 1 is shaded relative to Strings 2 and 3;

FIG. 6B shows simulation results for individual string voltages as functions of the total series current through the entire PV module shown in FIG. 4 while the voltage equalization circuit is operating and String 1 is shaded relative to Strings 2 and 3;

FIG. 6C shows simulation results for individual string output powers as functions of the total series current through the entire PV module shown in FIG. 4 while the voltage equalization circuit is operating and String 1 is shaded relative to Strings 2 and 3;

FIG. 7 shows an exemplary physical configuration in which one voltage equalization circuit is integrated in a junction box with each PV module and two conductors provide a connection between adjacent junction boxes;

FIG. 8 shows an extended configuration of the voltage equalization circuit shown in FIG. 4 in which the voltage equalization concept extends to series connected modules;

FIG. 9 shows a detailed schematic diagram of a voltage equalization circuit used in conjunction with a PV module according to an exemplary embodiment of the invention;

FIG. 10 shows an equivalent behavioral circuit of the voltage equalization circuit shown in FIG. 9;

FIG. 11A shows the configuration of the voltage equalization circuit shown in FIG. 9 during Phase 1 of its operation;

FIG. 11B shows the configuration of the voltage equalization circuit shown in FIG. 9 during Phase 2 of its operation;

FIG. 12 is a conceptual diagram illustrating the energy transfer and power flow of the voltage equalization circuit shown in FIG. 9 during Phases 1 and 2;

FIG. 13A shows an exemplary embodiment in which each of the energy storage devices in the voltage equalization circuit shown in FIG. 9 is a resonant device that includes a series L-C-R circuit;

FIG. 13B shows an energy storage device in which the inductor and the capacitor are implemented as discrete components on a printed circuit board;

FIG. 13C shows an energy storage device in which the inductor is implemented with a printed circuit board interconnect layer;

FIG. 13D shows an energy storage device in which only the capacitor is implemented as a discrete component on a printed circuit board;

FIG. 14 is a plot of the magnitude of the impedance of the series resonant L-C-R circuit shown in FIG. 13;

FIG. 15 shows the operation of the series resonant L-C-R circuit shown in FIG. 13;

FIG. 16 shows a schematic diagram in which the voltage equalization circuit shown in FIG. 9 is extended by adding an additional balancing stage;

FIG. 17 shows an example in which the extended voltage equalization circuit shown in FIG. 16 is used in conjunction with three PV modules connected in series;

FIG. 18 is a schematic diagram of a simplified network communication system that operates in conjunction with the extended voltage equalization circuit shown in FIG. 16;

FIG. 19 shows an example of the simplified network communication system shown in FIG. 18 in conjunction with three PV modules that are connected in series;

FIG. 20 shows an exemplary embodiment in which the single-pole, double-throw switches of the voltage equalization circuit are implemented with MOSFETs;

FIG. 21 shows an exemplary embodiment in which the energy storage devices of the voltage equalization circuit shown in FIG. 9 are implemented as capacitors;

FIG. 22A shows the configuration of the voltage equalization circuit shown in FIG. 21 during Phase 1 of its operation;

FIG. 22B shows the configuration of the voltage equalization circuit shown in FIG. 21 during Phase 2 of its operation;

FIG. 22C shows an equivalent behavioral circuit of the voltage equalization circuit shown in FIG. 21;

FIG. 23A shows simulated voltages across the strings of a PV module when one string is shaded and the voltage equalization circuit shown in FIG. 21 is active;

FIG. 23B shows simulated voltages across the un-shaded strings of the PV module when one string is shaded and the voltage equalization circuit shown in FIG. 21 is disabled;

FIG. 23C shows simulated voltages across the energy storage devices in the voltage equalization circuit shown in FIG. 21 when one string is shaded and the voltage equalization circuit is active;

FIG. 23D shows simulated voltages across the shaded string of the PV module when one string is shaded and the voltage equalization circuit shown in FIG. 21 is disabled;

FIG. 23E shows the clock when the voltage equalization circuit shown in FIG. 21 is active;

FIG. 24A is a plot that compares the peak power from the PV module with one string shaded when the voltage equalization circuit shown in FIG. 21 is active with the peak power from the PV module with one string shaded when the voltage equalization circuit is disabled;

FIG. 24B is a plot of the ratio between the peak power from the PV module with one string shaded when the voltage equalization circuit 400 is active and the peak power from the PV module with one string shaded when the voltage equalization circuit 400 is disabled;

FIG. 25A shows an equivalent R-C circuit for the capacitors in the voltage equalization circuit shown in FIG. 21;

FIG. 25B shows the charge-discharge cycle of the capacitors in the voltage equalization circuit shown in FIG. 21;

FIG. 26 shows a schematic diagram of a voltage equalization circuit used in conjunction with a PV module having six strings of PV cells according to an exemplary embodiment of the invention;

FIG. 27 shows a schematic diagram of a voltage equalization circuit used in conjunction with a string of six PV modules according to an exemplary embodiment of the invention;

FIG. 28 shows a schematic diagram of a voltage equalization circuit having separate circuits to balance the voltages of PV modules and strings within the PV module, according to an exemplary embodiment of the invention;

FIG. 29 shows a schematic diagram of another voltage equalization circuit having separate circuits to balance the voltages of PV modules and strings within the PV module, according to an exemplary embodiment of the invention;

FIG. 30 shows an exemplary embodiment that uses a single energy storage device to equalize voltages among multiple PV strings;

FIG. 31A shows the configuration of the voltage equalization circuit shown in FIG. 30 during Phase 1 of its operation;

FIG. 31B shows the configuration of the voltage equalization circuit shown in FIG. 30 during Phase 2 of its operation;

FIG. 31C shows the configuration of the voltage equalization circuit shown in FIG. 30 during Phase 3 of its operation;

FIG. 32A shows an exemplary embodiment that uses magnetic energy storage devices within the voltage equalization circuit; and

FIG. 32B shows an example of a circuit in which bi-directional (buck-boost) converters are used to create alternate current paths in parallel with the strings of a PV module.

DETAILED DESCRIPTION OF EXEMPLARY EMBODIMENTS

To address the problems discussed above, exemplary embodiments of the present invention manage the PV modules 20 with finer granularity. The power management method regulates smaller sections of the PV installation 10 to local optimum operating points, allowing the sections to operate substantially independently. The power management may occur at different levels, such as the level of the string of modules, the single module, the string of PV cells within a single module, or the single PV cell. Finer control allows sub-sections of the PV installation 10 to be shaded or mismatched without affecting the performance of the entire PV installation 10.

FIG. 2A shows a lumped circuit equivalent of a typical PV module 20. As shown in FIG. 2A, the PV module 20 includes strings 50 of series-connected PV cells 60. In FIG. 2A, each of the PV cells 60 is represented as a forward biased diode, and the incident photo-power is modeled as a local current source 70. FIG. 2B is a plot showing power and current as functions of voltage for a typical PV module 20. As shown in FIG. 2B, the PV module 20 has an MPP that occurs at the maximum product of voltage and current, which coincides with the product of the MPV and the MPC. Additional characteristics of the PV module 20 shown in FIG. 2B include the open circuit voltage VOC, which is the voltage across the terminals of the PV module 20 when zero current is flowing, and the short circuit current ISC, which is the current that flows out of the PV module when its output terminals are short circuited.

A PV module 20 typically has two or three strings 50 of series-connected PV cells 60. Each PV cell 60 typically provides 0.5 V to 0.9 V, so it is necessary to connect a number of PV cells 60 in series to achieve an output voltage in a range between 10 V and 50 V for the PV module 20.

As discussed above, the PV module 20 may suffer from shading. Here, shading is a general term for any effect that reduces the amount of incident photo-power reaching the PV cells 60. Such effects may result from physical shading by objects such as trees, or from debris, dust, or damage to the PV cells 60. Shading effects also result from mismatched installation of PV modules 20, such as PV modules 20 that are positioned on different sides of a roof or at different angles.

If a PV cell 60 is shaded, such as the PV cell 60 in the lower left-hand corner of FIG. 2A, the shading limits the current in the PV cell 60. The shaded PV cell 60 then limits the current available in all series-connected PV cells 60, such as the PV cells 60 within the left-most string 50 in FIG. 2A. Since many PV cells 60 may be connected in series, even small amounts of shading can significantly reduce the available power from the PV module 20.

Typically a bypass diode 80 is placed in parallel with each string 50 of PV cells 60 in the PV module 20. The bypass diode 80 provides an alternative path for current to flow if part or all of a string 50 is shaded. If a string 50 is substantially shaded, the current in the string 50 is limited, and the bypass diode 80 may turn on to short out the string 50. If the bypass diode 80 turns on, all power from the shaded string 50 is lost. Additional power is also lost in the bypass diode 80. Also, if the string 50 is bypassed, the voltage across the string 50 drops to the forward-biased diode potential.

Similarly, in a series string of PV modules 20, if one PV module 20 is shaded, such as the PV module 20 shown in the lower left-hand corner of FIG. 1, and the current in the series string of PV modules 20 is forced to the MPC, then bypass diodes 80 in the shaded PV module 20 may turn on to short out underperforming strings 50 of PV cells 60. Alternatively, the maximum current in the series string of PV modules 20 may be limited to the worst-case current in the shaded series string, significantly reducing the MPP of the series string.

FIG. 2C is a plot of the output power as a function of the output current for two mismatched PV modules 20 that are connected in series. The PV modules 20 may be mismatched if one of the PV modules 20 is shaded as discussed above. In the example shown in FIG. 2C, the second PV module 20 is shaded. FIG. 2C could also apply to two mismatched PV cells 60 or two mismatched strings 50 of PV cells 60. As shown in FIG. 2C, if the second PV module 20 is forced to operate with the MPC of the first PV module 20, the second PV module 20 is effectively shorted out by the bypass diode 80, and provides zero or negative power. The PV module 20 can provide negative power when the bypass diode 80 is on because the bypass diode 80 will consume power. In FIG. 2C the output power of the first PV module 20 is labeled as P1, and the output power of the second PV module 20 is labeled as P2.

In contrast, if the same two mismatched PV modules 20 are connected in parallel, their MPVs are often similar enough that minimal power loss occurs relative to their MPPs. This effect is shown in FIG. 2D, which is a plot of the output power as a function of the output voltage for the first PV module 20 and the second PV module 20. Again, the output power of the first PV module 20 is labeled as P1, and the output power of the second PV module 20 is labeled as P2. Therefore, to address the problems created by localized shading, it would be advantageous to modify the PV installation 10 such that the PV modules 20 are effectively connected in parallel instead of in series.

FIG. 3 shows a top-level schematic representation of a converter adapted to be used in conjunction with a PV module 20 in accordance with exemplary embodiments of the invention. The converter may include a voltage equalization circuit 100 and a system controller 110. In the present exemplary embodiment, the PV module 20 includes three strings 1, 2, and 3, each of which may include a plurality of PV cells 60 connected in series (not shown). As shown in FIG. 3, positive and negative voltage taps from each string 1, 2, and 3 of the PV module 20 are connected to the voltage equalization circuit 100. Outputs from the voltage equalization circuit 100 correspond to the typical positive terminal 120 and negative terminal 130 from the PV module 20, along with an auxiliary positive terminal 140 and an auxiliary negative terminal 150.

The voltage equalization circuit 100 may be controlled by the system controller 110. The system controller 110 may enable or disable the voltage equalization circuit 100. The system controller 110 may enable the voltage equalization circuit 100 if the system controller 110 detects that the strings 50 are unbalanced. As discussed above, the strings 50 may be unbalanced due to partial shading, dirt, debris, or damage. The system controller 110 may send an activation signal to the voltage equalization circuit 100 if the system controller 110 detects that one of the string voltages is substantially less than the average string voltage. In addition, the system controller 110 may measure the temperature of the PV module 20, the junction box 240, or the voltage equalization circuit 100 and disable the voltage equalization circuit 100 if one or more of the temperatures is outside of a threshold value.

As shown in FIG. 3, the inputs to the system controller 110 are the voltages VS1, VS2, and VS3 of the strings 1, 2, and 3 in the PV module 20, the overall PV voltage Vpv, the overall PV current Ipv, and the temperature Temp of the voltage equalization circuit 100. The system controller 110 also has an output voltage Vout.

FIG. 4 shows a conceptual overview of the operation of the exemplary embodiment shown in FIG. 3 in which the voltage equalization circuit 100 is connected to three strings 1, 2, and 3 of series-connected PV cells 60. This is a typical configuration for a single PV module 20 with three strings 1, 2, and 3 of PV cells 60. However, the voltage equalization concept may extend to more or less than three strings of series-connected PV cells.

The voltage equalization circuit 100 operates to force the voltages VS1, VS2, and VS3 across the series-connected strings 1, 2, and 3 to be substantially equal. In other words, the voltage equalization circuit 100 enforces the condition that VS1=VS2=VS3 is substantially satisfied. For example, each voltage VS1, VS2, and VS3 may be 30 V±2 V. The voltage equalization circuit 100 provides additional paths for current to flow in the strings 1, 2, and 3, as shown in FIG. 4 by the current paths 1, 2, and 3, respectively. Since all of the string voltages VS1, VS2, and VS3 are forced to be substantially equal, each string 1, 2, and 3 operates at the same voltage on the power-voltage curve, as shown in FIG. 2D. This is similar to the scenario in which the strings 1, 2, and 3 are connected in parallel, as schematically represented on the right-hand side of FIG. 4. If one string, such as string 1, is shaded, it does not limit the performance of the other strings connected in series, such as strings 2 and 3. Also, since the MPV typically does not change substantially between strings 1, 2, and 3, the shaded string 1 can continue to contribute power near its MPP. When the string voltages VS1, VS2, and VS3 are substantially equal, the bypass diodes 80 do not turn on to short out the strings 1, 2, and 3.

As shown in FIG. 5, the voltage equalization circuit 100 may extend to any number N of series-connected strings 1, 2, 3, . . . N of PV cells 60. Alternatively, each string could consist of a single PV cell 60, or each string could be replaced by a PV module that includes a plurality of series-connected strings. A power point tracking converter 200 with a load 210 may be used to identify the optimal MPP for the entire PV module 20 shown in FIG. 5. The power point tracking converter 200 may be an MPPT inverter or an MPPT DC-DC converter. The voltage equalization circuit 100 then enforces the correct MPP for each string 1, 2, 3, . . . N. Accordingly, each string 1, 2, 3, . . . N can operate near its MPP regardless of any shading. For example, if string 1 is shaded, it will continue to operate near its MPP without affecting the other series-connected strings 2, 3, . . . N.

An example of an MPPT algorithm is a hill-climbing algorithm, in which the power point tracking converter 200 steps a control signal in a continuous direction as long as the power continues to increase. The power point tracking converter 200 measures the PV voltage and current and calculates the PV output power. The power point tracking converter 200 steps the control signal up or down, and then measures the PV output power again. If the PV output power increases, the power point tracking converter 200 steps the control signal again in the same direction. If the PV output power decreases, the converter steps the control signal in the opposite direction. The control signal for an inverter or DC-DC converter may be a conversion ratio of the output voltage to the input voltage, the duty cycle, the PV voltage, the PV current, the output voltage, or the output current. The hill-climbing algorithm is given by:

$\begin{matrix} {{{{{Sigref}\left( {n + 1} \right)}T} = {{{{Sigref}(n)}T} + {{Sigstep}*{{SIGN}\left\lbrack \frac{\mathbb{d}P}{\mathbb{d}{{Sigref}(n)}} \right\rbrack}}}},} & (1) \end{matrix}$ where Sigref is the control signal for the MPPT algorithm, Sigstep is the magnitude of the hill-climbing step, P is the measured power, n is an iterative index, and SIGN is the sign function returning +1 or −1.

FIGS. 6A, 6B, and 6C show simulated current, voltage, and power curves for a PV module 20 with three series-connected strings 1, 2, and 3 and a voltage equalization circuit 100 as shown in FIG. 4. For this simulation, string 1 was shaded relative to strings 2 and 3. The x-axis is a sweep of the total series current Iout through the entire PV module 20. This could be the independent control variable of a MPP tracking load. For example, the power point tracking converter 200 shown in FIG. 5 could use Iout as the control signal to find the MPP by the method described above. In this case the power point tracking converter 200 adjusts the current Iout drawn from the entire PV module 20, while measuring both Iout and Vout of the PV module 20. The power drawn from the PV module 20 is the product of Iout and Vout. The power point tracking converter 200 modifies Iout while using an MPPT algorithm to drive the PV module 20 toward its MPP.

FIG. 6A shows the individual string currents versus Iout. As shown in FIG. 6A, the current through string 1 is lower than the current through strings 2 and 3, because string 1 is shaded. The additional current path 1 allows the current through string 1 to be lower. FIG. 6B shows the individual string voltages versus Iout. By using the voltage equalization circuit 100, the voltage across each of strings 1, 2, and 3 are forced to be substantially equal, regardless of the output current. FIG. 6C shows the individual string output powers versus Iout. Since string 1 is shaded, the peak power from string 1 is less than the peak power from the unshaded strings 2 and 3. However, the peak power point occurs at approximately the same Iout for each string 1, 2, and 3, due to the effects of the voltage equalization circuit 100. Therefore, the voltage equalization circuit 100 allows string 1 to operate with less current without affecting the conditions in strings 2 and 3.

As shown in FIG. 7, a typical PV module 20 may have between one and six strings 50 of PV cells 60, with parallel bypass diodes 80 integrated in a junction box (J-box) 240. According to exemplary embodiments of the present invention, the converter including the voltage equalization circuit 100 may also be integrated in the J-box 240 of the PV module 20. The connections for the converter from the J-box 240 are shown in FIGS. 3, 7, and 8.

As shown in FIGS. 3-5, the voltage equalization circuit 100 requires both positive and negative terminals of the strings 1, 2, and 3 to operate. In a single PV module 20 with three strings 1, 2, and 3, these terminals are available due to the typical wiring used to accommodate the bypass diodes 80. Therefore, to accomplish voltage equalization for strings 1, 2, and 3 in a single PV module 20, no modifications to the wiring of the PV module 20 are required.

However, to accomplish voltage equalization for a plurality of PV modules 20 connected in series, exemplary embodiments of the invention provide an auxiliary positive terminal 140 and an auxiliary negative terminal 150 for each of the PV modules 20, as shown in FIGS. 3 and 8. The auxiliary terminals 140 and 150 connect to adjacent PV modules via dual-conductor wires 250, as shown in FIGS. 7 and 8. The dual-connector wire 250 from PV module A is connected with the dual-connector wire 250 from PV module B via a dual-conductor connector 260. A voltage equalization circuit 100 is integrated with each of the PV modules A and B. As shown in FIG. 7, dual conductor wires 250 may be used to minimize the cost of the apparatus and simplify installation. Each dual conductor wire 250 includes two conductors that are co-sheathed in a single insulator. Alternatively, the J-boxes 240 could be connected with two single conductor wires.

For example, the additional dual-conductor wires 250 provide the potentials of strings 50 within PV module A to the adjacent series-connected PV module B. This allows a voltage equalization circuit 100 in PV module A to equalize all of the strings 50 in PV module A with one string 50 in PV module B. As shown in FIG. 8, the string voltages in PV module A are VS1, VS2, and VS3, and the string voltages in PV module B are VS4, VS5, and VS6. Also, PV module A receives VS4 from PV module B. The voltage equalization constraint in PV module A ensures that VS1=VS2=VS3=VS4, and the voltage equalization constraint in PV module B ensures that VS4=VS5=VS6, such that VS1=VS2=VS3=VS4=VS5=VS6.

With these constraints, the system ensures that all strings 50 within the series-connected PV modules A and B have substantially equal voltages. This concept can be extended to any number of PV modules in series that have any number of internal strings 50 of PV cells 60.

FIG. 9 shows a detailed schematic diagram of the voltage equalization circuit 100 in conjunction with a PV module 20 according to an exemplary embodiment of the invention. In the present embodiment, the voltage equalization circuit 100 works by dynamically modulating the configuration of energy storage devices Zx1 and Zx2 among series-connected PV strings 1, 2, and 3. The energy storage devices Zx1 and Zx2 store energy as a function of string voltage. Exemplary energy storage devices Zx1 and Zx2 include capacitors, inductors, batteries, fuel cells, piezoelectric devices, and mechanical energy storage devices including MEMS devices. Additional examples of energy storage devices Zx1 and Zx2 include networks that consist of capacitors, inductors, or other devices. Such networks may have a frequency-dependent impedance with resonant behavior. As discussed below, an example resonant network is a series-connected inductor and capacitor.

An equivalent behavioral circuit 190 that represents the behavioral action of the voltage equalization circuit 100 is shown on the right-hand side of FIG. 10. Here VS1, VS2, and VS3 are the voltages across the strings 1, 2, and 3 of the PV module 20, respectively. ZS1, ZS2, and ZS3 represent the nonlinear impedance of the strings 1, 2, and 3, respectively, as could be derived from the curves shown in FIGS. 2B-2D.

The quantity Reff is an effective resistance that is created by the voltage equalization circuit 100 shown in FIGS. 9 and 10. The equivalent behavioral circuit 190 of the voltage equalization circuit 100 shown in FIG. 10 includes effective voltage sources VS1, VS2, and VS3 that are shown in series with Reff. These effective voltage sources VS1, VS2, and VS3 and Reff are modeled in parallel with terminals connecting to the strings 1, 2, and 3. For example, VS1 in string 1 is placed in parallel with VS2 from string 2 in series with Reff. If Reff is substantially small relative to impedances ZS1-ZS3, then voltages VS1-VS3 will be forced substantially equal. The equalization process happens due to the feedback inherent in the behavioral circuit of FIG. 9. In practice, Reff may be on the order of 100 mΩ, and the real part of ZS1-ZS3 may be on the order of several ohms.

To first-order, power loss in the voltage equalization circuit 100 is related to the equivalent resistance Reff. Assuming that the differences among voltages VS1, VS2, and VS3 are small, the differences in current among the strings 1, 2, and 3 will flow through the voltage equalization circuit 100. Power loss will follow according to P_(loss-branch)=I_(Branch) ²·R_(branch) where I_(Branch) is the current in a branch of the voltage equalization circuit 100 and R_(Branch) is the equivalent resistance of that branch. Here R_(Branch) may be equal to Reff. The power loss may also be expressed as P_(loss-branch)=V_(Branch) ²/R_(branch), where V_(Branch) is the voltage difference between adjacent strings of PV cells 60. A smaller Reff will result in a smaller V_(Branch) and a lower power loss. Therefore it may be advantageous to design the voltage equalization circuit 100 to reduce Reff as much as possible.

As shown in FIGS. 11A and 11B, the voltage equalization circuit 100 operates in two phases controlled by a switch signal at a frequency f₀. Phase 1 is shown in FIG. 11A and Phase 2 is shown in FIG. 11B. The switch signal drives the control terminal, VG, of each of the single-pole, double-throw switches S1 b, S2 b, and S3 b shown in FIGS. 9 and 10. The voltage equalization circuit 100 operates with a substantially 50% duty cycle, spending approximately half the time in Phase 1 and half the time in Phase 2.

In Phase 1, the first energy storage device Zx1 is configured in parallel with VS2. At the end of phase 1, the first energy storage device Zx1 is storing energy E1. In phase 2, the first energy storage device Zx1 is configured in parallel with VS1. At the end of phase 2, the second energy storage device Zx2 is storing energy E2. If E1 is greater than E2, some energy is transferred from VS2 to VS1. A corresponding process occurs for the second energy storage device Zx2 in Phases 1 and 2. Assuming that energy ΔE=E1−E2 is transferred from VS2 to VS1 in every switching cycle with a period

${T_{0} = \frac{1}{f_{0}}},$ power P=f₀ΔE is transferred on average from VS2 to VS1. FIG. 12 is a conceptual diagram illustrating this operation of the voltage equalization circuit 100.

Since power flows in the direction from a higher voltage to a lower voltage, current also flows from the higher voltage to the lower voltage. This current flow will act against the impedance of each voltage tap to increase the voltage of taps that are low and decrease the voltage of taps that are high. When the voltages across taps are equal, no average current flow occurs and the energy transfer approaches zero.

Although FIGS. 11A and 11B show an embodiment in which the energy storage devices Zx1 and Zx2 are reconfigured between Phases 1 and 2 to be connected in parallel with adjacent strings of the PV module 20, the energy storage devices Zx1 and Zx2 could also be reconfigured between Phases 1 and 2 to be connected in parallel with any of the strings within the PV module 20. Each energy storage device Zx1 and Zx2 merely needs to be connected across a different string within the PV module in Phases 1 and 2. The present invention is not limited by the geometry shown in FIGS. 11A and 11B.

In an exemplary embodiment of the invention, each of the energy storage devices Zx1 and Zx2 is a resonant device that includes a series resonant L-C-R circuit with an inductor L_(R) and a capacitor C_(R) connected in series, as shown in FIG. 13A. The L-C-R circuit also includes an effective series resistance R₀. The effective series resistance R₀ results from parasitic resistance in the inductor L_(R) and the capacitor C_(R). In practice the effective series resistance R₀ also includes any resistance in the switches S1 b, S2 b, and S3 b and the bypass capacitors Cbp. In this embodiment, energy stored in each of the energy storage devices Zx1 and Zx2 is a function of frequency and voltage.

The inductor L_(R) schematically depicted in FIG. 13A lumps together all inductance in series with the capacitor C_(R) in the circuit. The inductor L_(R) can include any inductance that is intentionally connected in series with the capacitor C_(R), any parasitic inductance from circuit interconnect, and any parasitic inductance in the capacitor C_(R). Inductance inherent in the capacitor C_(R) is traditionally called the effective series inductance (ESL) of the capacitor C_(R). The ESL of the capacitor C_(R) results in a self-resonant frequency of the capacitor C_(R) that occurs at a frequency

${f_{SR} = \frac{1}{2\pi\sqrt{L_{ESL}C_{R}}}},$ where L_(ESL) is the ESL of the capacitor C_(R).

FIG. 13B shows an embodiment of an energy storage device in which the inductor L_(R) and the capacitor C_(R) are implemented as discrete components on a printed circuit board (PCB) 220. The inductor L_(R) and/or the capacitor C_(R) may be implemented with a single device, or with multiple devices connected to the PCB 220. In FIG. 13B, the switches S1 b and S2 b shown in FIGS. 9 and 10 are implemented with discrete switches S1, S2, S3, and S4, respectively.

FIG. 13C shows an embodiment of an energy storage device in which the inductor L_(R) is implemented with interconnect layers 230 of the PCB 220. Inductance can be created with interconnect layers 230, which are planar metallization layers in the PCB 220, by designing loops that link magnetic flux. The inductance of a planar circular spiral is approximately L≈4π×10⁻⁷ n²r, where L is the inductance of the spiral, n is the number of turns of the spiral, and r is the radius of the spiral in meters. In practice, the inductor L_(R) includes any inductance from any loops in series with the capacitor C_(R) that link magnetic flux, including any unintentional loop area from metallization that connects the capacitor C_(R), the switches S1 b and S2 b, and the bypass capacitors Cbp. In the present embodiment, the inductor L_(R) also includes the ESL of the capacitor C_(R).

FIG. 13D shows an alternative embodiment of a conceptual PCB 220 with no inductance intentionally placed in series with the capacitor C_(R). Here, the inductor L_(R) of the L-C-R circuit can include the ESL of the capacitor C_(R) and any stray inductance from the interconnect layers 230 of the PCB. The switching process in this circuit may occur at substantially the self-resonant frequency of the capacitor C_(R). Alternatively, the inductor L_(R) could be treated as having negligible inductance. In this case the energy storage device would consist only of the capacitor C_(R). In FIGS. 13B-13D the PCB 220 may be connected to an integrated circuit (not shown).

FIG. 14 is a plot of the impedance of the series resonant L-C-R circuit shown in FIG. 13. As shown in FIG. 14, the impedance of the L-C-R circuit is dominated by the capacitor C_(R) at low frequencies, and by the inductor L_(R) at high frequencies. The resonant frequency f₀ occurs when the magnitudes of the impedance of the inductor C_(R) and capacitor L_(R) are the same, but 180° out of phase. At resonance, the impedances of the inductor L_(R) and the capacitor C_(R) cancel each other, and the net impedance of the network is the effective series resistance R₀. This occurs at frequency

$\begin{matrix} {f_{0} = \frac{1}{2\pi\sqrt{L_{R}C_{R}}}} & (2) \end{matrix}$ which is the resonant frequency f₀ of the L-C-R circuit.

An important characteristic of the resonant L-C-R circuit is the quality factor Q of the circuit. The quality factor Q is defined as the ratio of energy stored to energy dissipated per cycle. The quality factor Q of the L-C-R circuit is given by:

$\begin{matrix} {Q = {\frac{1}{R_{0}}\sqrt{\frac{L_{R}}{C_{R}}}}} & (3) \end{matrix}$ such that a smaller effective series resistance R₀ leads to a higher quality factor Q. For Q>>1, the current in the circuit is approximately sinusoidal for a voltage stimulus that is periodic at the resonant frequency f₀.

As shown in FIG. 15, the series resonant voltage equalization circuit 100 works by transferring energy between two voltage taps V1 and V2. A control signal VG modulates the configuration of the switch such that Zx is in parallel with V1 in phase 1, and in parallel with V2 in phase 2. The control signal VG operates at the resonant frequency f₀ of the L-C-R circuit with a substantially 50% duty cycle. Vx is the voltage across the resonant impedance, and Ix is the current through the resonant impedance. The waveforms shown in FIG. 15 are for the steady state operation of the L-C-R circuit.

In phase 1, Zx is in parallel with V1. Since V1 is larger than V2, the current Ix in the resonant impedance follows the positive half-wave sinusoidal trajectory. The current Ix reaches a peak is at ¼ of the total period. At the end of phase 1, the current Ix in the circuit approaches zero. The control signal VG switches at the end of phase one at ½ of the period when the current Ix is approximately zero. During phase 1, Zx draws net energy from V1.

In phase 2, Zx is in parallel with V2. Since V2 is less than V1, the current Ix in the resonant impedance follows the negative half-wave sinusoidal trajectory. The current Ix reaches a minimum −ia at ¾ of the total period. During phase 2, Zx delivers energy to V2.

Since the L-R-C circuit operates at the resonant frequency f₀, the current Ix in the circuit is constrained by the effective series resistance R₀ and the difference in voltage V1−V2. With a quality factor Q substantially larger than one, the current magnitude ia is

$\begin{matrix} {{ia} = {\frac{\frac{2}{\pi}\left( {{V\; 1} - {V\; 2}} \right)}{R_{0}}.}} & (4) \end{matrix}$ The average current transferred between V1 and V2 is

$\begin{matrix} {{\overset{\_}{idc} = {\frac{ia}{T_{0}}{\int_{0}^{\frac{T_{0}}{2}}{{\sin\left( {\frac{2\pi}{T_{0}}t} \right)}\ {\mathbb{d}t}}}}},} & (5) \end{matrix}$ which results in

$\begin{matrix} {\overset{\_}{idc} = {\frac{ia}{\pi} = {\frac{\frac{2}{\pi^{2}}\left( {{V\; 1} - {V\; 2}} \right)}{R_{0}}.}}} & (6) \end{matrix}$ With an effective resistance defined as

${R_{eff} = \frac{{V\; 1} - {V\; 2}}{\overset{\_}{idc}}},$ it follows that

$\begin{matrix} {R_{eff} = {{R_{0}\frac{\pi^{2}}{2}} \approx {5 \times {R_{0}.}}}} & (7) \end{matrix}$ This expression for Reff is used along with FIG. 10 to determine the steady state operation of the voltage equalization circuit 100.

Although the above discussion describes a switching process that occurs at the resonant frequency f₀ of the L-R-C circuit with a 50% duty cycle, other switching frequencies and duty cycles are possible. The switching process can also operate at frequencies less than or greater than the resonant frequency f₀. The circuit may be most effective when an odd harmonic of the switching process is substantially the same as the resonant frequency f₀. For example, if the resonant frequency f₀ of the L-R-C circuit is at 1 MHz, the switching process may operate at substantially 1 MHz, 333 kHz, or 200 kHz, such that the 1^(st), 3^(rd), or 5^(th) harmonic, respectively, is substantially the same as the resonant frequency f₀.

In order to extend the voltage equalization circuit 100 to additional PV modules 20, one or more strings 50 of a PV module may be balanced with one or more strings 50 of an adjacent PV module 20. This extension of the voltage equalization circuit 100 may be achieved by adding the additional balancing stage 300 shown FIG. 16. This additional balancing stage 300 includes an additional bypass capacitor Cbp4, an additional switch S4 b, and an additional balancing or flying impedance Zx3.

As shown in FIG. 16, two additional terminals are used to provide positive and negative voltages of string 1 to the adjacent PV module 20. The terminals out of the package or junction box are the typical positive terminal TB and negative terminal TA of the PV module 20, along with a positive terminal TC of string 1 of the PV module 20, and a terminal TD to accept the positive voltage of string 1 of the adjacent PV module 20. Terminals TA and TC can be housed in the same connector out of the package or junction box. Similarly, terminals TB and TD can be housed in the same connector out of the package or junction box. Wires A-D pass through terminals TA-TD to connect to adjacent PV modules 20 or DC bus connectors. Wires B and D can be housed in the same insulator or sheath and can use a single connector at each package or junction box interface terminal. This helps to minimize the cost of wiring material and total number of connectors. Wires A and C can similarly be housed in the same sheath and use single connectors at each terminal. Alternatively, wires A and C can be mechanically connected in parallel to minimize the number of free wires in the system. Similarly, wires B and D can be mechanically connected in parallel to minimize the number of free wires in the system.

FIG. 17 shows an example in which three PV modules 1, 2, and 3 are connected in series with the extended circuit shown in FIG. 16. Connections between the PV modules 1, 2, and 3 are provided by wires A-H. As shown in FIG. 17, string 1 of PV module 2 is balanced with string 3 of PV module 1. Further, string 1 of PV module 3 is balanced with string 3 of PV module 2. In the present embodiment, the centralized inverter 30 may find the MPP for the entire array of PV modules 1, 2, and 3. Since all of the series-connected strings within the PV modules 1, 2, and 3 are forced to have substantially equal voltages, each string operates at substantially the MPP as determined by the centralized inverter 30.

FIG. 18 shows an example of a simplified network communication system in conjunction with the extended circuit shown in FIG. 16. FIG. 18 shows that each package or junction box associated with the PV module 1 has a communications circuit 310 that communicates information to adjacent PV modules and back to the centralized inverter 30. Communications are simplified because two wires, such as wires A and B or wires C and D, are routed from each PV module to adjacent PV modules. These two wires are used to perform wireline communications based on the voltage potential difference between the two wires. Wireline communication does not interfere with power flow among the PV modules or the centralized inverter 30 because the signals have a low voltage amplitude and a high frequency relative to the DC power.

Each communications circuit 310 may have one or more transmitters TX, one or more receivers RX, and a communications controller 320 that generates modulated signals and can process received signals. The communications controller 320 can read data from adjacent PV modules over the wireline link and re-transmit the data to other adjacent PV modules, thereby forming a daisy-chain communications link. The communications block 330 at the end of the series string of PV modules can communicate directly to a wireline communications block at the centralized inverter 30 or at some other location in the system. The transmitter and receiver in this other communications block (not shown) is similar to the communications block 330 shown in FIG. 18. The other transceiver (not shown) can be associated with the centralized inverter 30, but may also be in any other location in the system as long as it has access to two wires, such as wire A and wire B.

Alternatively, the wireline communications signal may be transmitted as current instead of voltage potential. The communications signal may be a digital (binary 1 or 0) type signal (a baseband signal), or it may be a modulated carrier signal with a frequency tone that is subject to amplitude, phase, or frequency modulation.

FIG. 19 shows an example of the simplified network communication system in which three PV modules 1, 2, and 3 are connected in series. As shown in FIG. 19, PV module 1 communicates with the centralized inverter 30 and/or PV module 2. PV module 2 communicates with PV module 1 and/or PV module 3. PV module 3 communicates with PV module 2 and/or the centralized inverter 30. The centralized inverter 30 communicates with PV module 1 and/or PV module 3. However, the communication system is not necessarily associated with the centralized inverter 30. Instead, there may be a separate data logging, fault detection, or programming device.

In an exemplary embodiment of the present invention, the single-pole, double-throw switches SNb may be implemented with MOSFETs, as shown in FIG. 20. For example, a switch SNb may be implemented with two MOSFETs S1 and S2. Instead of a single control signal VG, the MOSFETs S1 and S2 are controlled by voltage potentials Va and Vb, respectively. MOSFETs are typically three-terminal devices with gate, drain, and source terminals. The voltage potential across the gate-source terminal controls the current flow between the drain and the source.

As shown in FIG. 20, Va is synchronous with VG, forcing Vx substantially equal to V1 when Va=Vx+Vdrive. Vx is the source potential of S1 and Vdrive is the excess potential required to substantially turn on S1. Vdrive may exceed the threshold voltage of S1 by some amount to substantially reduce the resistance of S1 when S1 is on. When S1 is off, Va is substantially equal to Vx, forcing the gate-source voltage to substantially zero. Va may also be substantially less than Vx to ensure that S1 is off with a margin of safety.

Similarly, S2 is controlled by Vb. When S2 is off, Vb is substantially equal to V2 forcing the gate-source potential to substantially zero. S2 turns on only when S1 has turned off. To prevent S1 and S2 from turning on simultaneously, a timing delay is included between the on-off periods of the two switches S1 and S2. As shown in FIG. 20, this timing delay may be known as deadtime DT, which is the time when both S1 and S2 are off When S2 turns on, Vb is driven to potential V2+Vdrive. Vdrive may be substantially higher than the threshold voltage of S2 to substantially reduce the resistance of S2 during the on period.

Although S1 and S2 are described here as N-channel MOSFETs, they may also be implemented with other technologies. For example, S1 and S2 could be implemented as P-channel MOSFETs by changing the polarity of the gate to source drive potential to negative. S1 and S2 could also be implemented as bipolar transistors, insulated-gate bipolar transistors (IGBTs), relays, junction field effect transistors (JFets), or other switch technologies.

As shown in FIG. 21, the energy storage devices of the voltage equalization circuit may be implemented as capacitors in another exemplary embodiment of the invention. FIG. 21 is similar to FIG. 9, but replaces the energy storage devices Zx1 and Zx2 with capacitors Cx1 and Cx2. Capacitors provide an example of a simple non-resonant energy storage device. Capacitors store energy as a function of voltage:

$\begin{matrix} {E = {\frac{1}{2}{CV}^{2}}} & (8) \end{matrix}$ where E is energy stored in the device, C is the capacitance, and V is the voltage across the device.

The circuit shown in FIG. 21 operates as the reverse of typical battery equalization circuits, because current flows out of PV modules 20, rather than flowing in as is the case with battery charging. Such circuits route current among differential voltage taps in a series string in order to force the voltages to be substantially equal. This is effective if the differential voltages have some positive nonzero impedance or resistance, such as the effective series resistance (ESR) of the battery.

The voltage equalization circuit 400 shown in FIG. 21 works by transferring charge between two differential voltage taps. As shown in FIGS. 22A and 22B, the voltage equalization circuit 400 operates in two phases controlled by a switch signal at a frequency f₀. Phase 1 is shown in FIG. 22A and Phase 2 is shown in FIG. 22B. In Phase 1 the capacitors Cx1 and Cx2 are initially connected across voltages VS3 and VS2, respectively. In Phase 1 the capacitors Cx1 and Cx2 hold charge equivalent to Cx1*VS3 and Cx2*VS2, respectively. In Phase 2 the capacitors Cx1 and Cx2 are connected across VS2 and VS1, respectively. Eventually the capacitors Cx1 and Cx2 hold charge Cx1*VS2 and Cx2*VS1, respectively. The difference in charge (Cx1*VS3−Cx1*VS2) will flow into or out of the respective bypass capacitor Cbp. If VS3 is larger than VS2, then charge will flow from VS3 to VS2 and vice versa. This sets up a feedback mechanism in which if any one of VS3, VS2, or VS1 is larger than the adjacent voltage, charge will flow to equalize the voltages.

Charge is transferred at a frequency fclk. Therefore the currents among the nodes will be Qi*fclk, where Qi is the charge in the node. Switched capacitor circuits are commonly analyzed with equivalent resistances Ri=1/(Ci*fclk). Similar analysis applies here where equivalent resistances are connected among the differential voltages with magnitude R2=1/(Cx2*fclk).

FIG. 22C shows an equivalent behavioral circuit 410 of the voltage equalization circuit 400 with an effective resistance R=1/(Cx*fclk). Here voltage sources V1-V3 represent an effective voltage from the PV module 20. Z1-Z3 represent the nonlinear impedance of the PV module 20, as could be derived from the curves in FIGS. 2B-D. VS1-VS3 are the voltages across the strings 1-3 in the PV module 20. With an equivalent resistance R substantially less than Z1-Z3, the string voltages VS1-VS3 will be made to be substantially equal.

First-order power loss in the voltage equalization circuit 400 is related to the equivalent resistance R. Assuming that the differences between the voltages VS1, VS2, and VS3 are small, the differences in current among the strings 1, 2, and 3 will flow through the voltage equalization circuit 400. Power loss will follow P_(loss-branch)=I_(Branch) ²·R_(branch), where I_(Branch) is the current in a branch of the voltage equalization circuit 400 and R_(Branch) is the equivalent resistance of that branch. Power loss is also expressed as P_(loss-branch)=V_(Branch) ²/R_(branch), where V_(Branch) is the voltage difference between adjacent strings 1, 2, and 3.

In general, a smaller equivalent resistance R will result in a lower power loss. Therefore a higher switching frequency fclk and larger capacitors Cx1 and Cx2 will reduce the power loss. The capacitance of the capacitors Cx1 and Cx2 is generally limited by the physical size, cost, or increasing parasitic elements such as resistance and ground-referenced capacitance. The switching frequency fclk is limited by power losses that are proportional to the switching frequency fclk, such as the power to drive the control terminal of switches S1 b-S3 b and charging ground-referenced capacitance connected to the nodes of the capacitors Cx1 and Cx2. The switching frequency fclk is also limited by the time constant of the R-C circuit formed by the relevant switch and capacitor, as described in next section.

The voltage equalization circuit 400 may be designed to have an equivalent resistance R in the range of 1.0Ω to 0.1Ω with Cx1=Cx2=10 μF and fclk in the range of 100 kHz to 1 MHz. The first order power loss in the system is inversely proportional to the equivalent resistance R. In some cases the switching frequency fclk may be increased when the differences among VS1, VS2, and VS3 increase. Increasing the switching frequency fclk will reduce the equivalent resistance R and the associated power loss with the voltage equalization circuit 400.

FIGS. 23A-E show simulated results from the PV module 20 and the voltage equalization circuit 400 shown in FIG. 21 using a spice simulator. During the simulation, string 1 is intentionally mismatched by setting the short circuit current ISC of string 1 to 6.3 A. The other two strings 2 and 3 each have a short circuit current ISC of 8.3 A. The PV module 20 is then forced to its MPP by the voltage equalization circuit 400.

FIG. 23A shows the voltages VS1, VS2, and VS3 across the strings 1, 2, and 3 when the voltage equalization circuit 400 is active, while FIG. 23B shows the voltages VS2 and VS3 across the un-shaded strings 2, and 3 when the voltage equalization circuit 400 is disabled. As shown in FIG. 23B, when the voltage equalization circuit 400 is disabled, string 1 is shorted out by the bypass diode 80, such that string 1 contributes no power. However, when the voltage equalization circuit 400 is active, current is routed around the underperforming string 1, enabling string 1 to deliver power near its MPP. FIG. 23C shows voltages across the energy storage devices Vx1 and Vx2 when they are configured in parallel with the different strings in Phase 1 and Phase 2. FIG. 23D shows the voltage VS1 across the shaded string 1 of the PV module 20 when the voltage equalization circuit 400 is disabled. FIG. 23E shows the clock as the voltage equalization circuit 400 alternates between Phase 1 and Phase 2.

FIG. 24A compares the peak power from the PV module 20 shown in FIG. 21 when one string is shaded and the voltage equalization circuit 400 is active with the peak power from the PV module when one string is shaded the voltage equalization circuit 400 is disabled. The x-axis is the short circuit current ISC of the shaded string, which is proportional to the amount of shading of the shaded string. As shown in FIG. 24A, when one string is shaded such that its short circuit current ISC is reduced to 6.5 A and the voltage equalization circuit 400 is active, a peak power of 205 W is produced by the PV module 20. In contrast, when one string is shaded and the voltage equalization circuit 400 is disabled, a peak power of 140 W is produced by the PV module 20.

FIG. 24B plots the ratio of the peak power from a single PV module when one string is shaded and the voltage equalization circuit 400 is active with the peak power from the PV module when one string is shaded and the voltage equalization circuit 400 is disabled. Here even modest shading is shown to result in substantial power savings when the voltage equalization circuit 400 is enabled. FIG. 24B shows that the benefit ratio is substantially 1.4 in this scenario, as approximately 40% more power is available when the voltage equalization circuit 400 is activated. As shown in FIG. 24B, the beneficial effects of the voltage equalization circuit 400 are highest in modest shading scenarios when significant power can be recovered from strings that are only slightly shaded, but would otherwise be shorted out by bypass diodes 80 because their short circuit current ISC is below the MPC of the PV module 20 of series-connected strings 1, 2, and 3.

The majority of power loss in the voltage equalization circuit 400 occurs when charge is transferred into and out of the capacitors Cx1 and Cx2. As is typical in switched capacitor circuits and charge pumps, the power loss is proportional to the difference in voltage before and after the capacitor Cx1 or Cx2 is charged. FIG. 25A shows an equivalent R-C circuit for the capacitors Cx1 and Cx2, each of which is represented by Cx. FIG. 25B shows the charge-discharge cycle of the capacitors Cx1 and Cx2. The effective resistance of switches S1 a-S1 c plus the effective series resistance (ESR) of Cx1 and Cx2 is modeled by R_(sw-esr). V1 and V2 are held constant in this example. As shown in FIG. 23B, when the gate voltage VG is high, the switch connects Cx from V1 to V2. The node Vx increases with the RC time constant τ_(Cx)=R_(sw-esr)·Cx from V1 to V2. The time constant τ_(Cx) sets a constraint on the switching frequency fclk. The node Vx must become substantially equal to V2 before the switch again connects Cx to V1. The timing constraint is set such that:

$\begin{matrix} {{\frac{T_{clk}}{2} > {n \cdot \tau_{Cx}}},} & (9) \end{matrix}$ where n is typically substantially larger than 3. This sets a maximum switching frequency of fclk<1/(2·n·τ_(Cx)).

Energy loss in the circuit occurs during every switching transition. As is well known with R-C circuits, the energy loss is not dependent on the effective resistance R. Instead, the energy loss in the circuit of FIGS. 25A and 25B is given by E_(lost)=Cx·(V2−V1)² such that power loss is P_(loss)=E_(lost)·fclk. Referring to FIGS. 21 and 22A-22C, assuming that VS3 and VS2 are substantially constant, the power loss associated with the capacitor Cx1 is P _(Cx1) =Cx1*(VS3−VS2)² *fclk,  (10) which is commonly referred to as CV² loss. Similar loss mechanisms are associated with the capacitor Cx2. In the nominal case, the loss is not a function of the resistance of switches S1 a-S1 c, or the ESR of capacitors Cx1 and Cx2.

In the previous description and in FIG. 23, it is assumed that V1 and V2 (VS3 and VS2) are substantially different and constant. However, the switched capacitor circuit will operate to force these voltages substantially equal. If the voltages V1 and V2 are substantially equal, the power loss in the circuit will approach zero. Using the terminology in FIG. 22C, with a smaller effective resistance R, VS3 will be closer in magnitude to VS2. In FIGS. 23A-E, this approaches the case that V1=V2. In this case the power loss approaches zero. Therefore it is typically desirable to achieve low effective resistance R in the switched capacitor circuit to achieve high power efficiency. A low effective resistance R is achieved with a high switching frequency fclk and high capacitance of the capacitors Cx1 and Cx2. Limits on the effective resistance R are dictated by the switching frequency fclk and the size of available capacitors Cx1 and Cx2. Limits on the switching frequency fclk are dictated by the size of the capacitors Cx1 and Cx2, R_(sw-esr), and the power loss associated with turning on and off the switches S1 a-S1 c.

For example, the resistance of the switches S1 a-S1 c and ESR of the capacitors Cx1 and Cx2 must permit the voltage across the capacitors Cx1 and Cx2 to substantially reach a steady state in one half of the switching period

$\left( \frac{1}{2*{fclk}} \right).$ This places a constraint on the time constant of the R-C circuit formed. Therefore, the switch resistance may be chosen to be substantially less than 15 mΩ, such that the circuit can switch at switch frequencies fclk as high as 1 MHz with the capacitance of the capacitor Cx1 as high as 10 μF.

In other exemplary embodiments of the invention, voltages may be equalized for more or less than 3 strings in a PV module 20. The voltage equalization concept may extend to any number of strings 50 having any number of PV cells 60. For example, FIG. 26 shows a voltage equalization circuit 450 in conjunction with a PV module 460 having 6 strings 1-6. For N strings there will be N+1 terminals accessible from the PV module 460 (typically in the junction box connector). The circuit will require N switches S1 b-SNb and N−1 capacitors Cx1-Cx(N−1). Similar to the embodiment shown in FIG. 21, the embodiment shown in FIG. 26 operates by forcing the string voltages VS1-VS6 to be substantially equal.

Alternatively, as shown in FIG. 27, the voltage equalization concept may operate on a string 470 of PV modules 1-6 to balance the voltages at the output of each PV module 1-6. Here each PV module 1-6 includes one or more strings 50 of PV cells 60 in series and/or parallel configurations. The voltage equalization circuit 480 operates to force the voltages VM1-VM6 substantially equal. Accordingly, the string 470 of PV modules 1-6 can provide more power to a load 210 when operated at the MPP of the string 470, especially when one or more of the PV modules 1-6 is shaded, causing the local MPCs to be substantially different.

Further, as shown in FIG. 28, the voltage equalization concept may be used to balance the voltages of strings 1, 2, and 3 within each PV module 1 and 2, and also to balance the voltages of the PV modules 1 and 2. Here, the voltage equalization circuit 510 includes a first circuit 520 and a second circuit 530 for separately equalizing the voltages of the strings 1, 2, and 3 within the PV modules 1 and 2, respectively. Each of the first circuit 520 and the second circuit 530 is similar to the voltage equalization circuit 400 shown in FIG. 21. The first circuit 520 and the second circuit 530 may operate independently at different switch frequencies fclk1 and fclk2. The voltage equalization circuit 510 also includes a third circuit 540 that equalizes all of the voltages within the PV modules 1 and 2. The first circuit 520 and the second circuit 530 may operate at substantially higher or lower frequencies than the third circuit 540.

It may be practical to integrate some or all of the third circuit 540 with the first circuit 520 or the second circuit 530 of any of the individual PV modules 1 and 2. In this case at least one additional wire may be necessary to connect between the PV modules 1 and 2 to provide both the positive and negative terminals of each PV module to the adjacent PV module.

FIG. 29 shows another exemplary embodiment in which the third circuit 540 of FIG. 28 is replaced with a fourth circuit 550. In FIG. 29 string 3 of PV module 1 is balanced with string 1 of PV module 2. The fourth circuit 550 requires only one additional capacitor CM1. In a practical implementation, CM1 could be integrated with the first circuit 520 or the second circuit 530. In this embodiment, an additional wire would be required to connect PV module 1 with PV module 2.

FIG. 30 shows another exemplary embodiment in which the voltage equalization circuit 580 uses a single energy storage device Zx1. The energy storage device Zx1 is reconfigured among parallel voltage taps VS1, VS2, and VS3 to equalize the voltages VS1, VS2, and VS3 across strings 1, 2, and 3, respectively. In this embodiment, switches S1 b and S2 b are single pole, triple throw switches that are controlled by a signal VG. The energy storage device Zx1 may be a resonant impedance circuit, a capacitor, or any other energy storage device that stores energy as a function of voltage.

As shown in FIGS. 31A-C, the voltage equalization circuit 580 operates in three phases such that the energy storage device Zx1 is configured in parallel with each of VS1, VS2, and VS3 in a single clock period. FIGS. 31A, 31B, and 31C show the configuration of the voltage equalization circuit 580 in Phase 1, Phase 2, and Phase 3, respectively. The configuration shown in FIG. 30 can be used to equalize the voltages across any number of strings. When only a single energy storage device Zx1 is used, the number of phases equals the number of strings to be equalized.

FIG. 32A shows another exemplary embodiment that uses magnetic-based DC-DC converter elements within the voltage equalization circuit 600. In the general case the DC-DC converters are represented as current sources 11, 12, and 13 in parallel with the strings 1, 2, and 3. The currents IA, IB, and IC represent the currents generated by the DC-DC converters that flow between nodes of the strings 1, 2, and 3.

In the example shown in FIG. 32B, bi-directional (buck-boost) converters 620 are used to create alternate current paths in parallel with the strings 1, 2, and 3. Here Ix and Iy are the currents delivered to the intermediate string nodes. Switches are controlled with signals Vg1 and Vg2 with pulse-width-modulation duty cycles d1 and d2. In steady state, the voltages across each of the strings 1, 2, and 3 are substantially equal (VS1=VS2=VS3), such that d1=d2=0.5 (the duty cycles are substantially equal to 50%). The nodal current constraints result in the following steady state equations for Ix and Iy:

$\begin{matrix} {{Ix} = {\frac{4}{3}\left\lbrack {\frac{{IB} + {IC}}{2} - {IA}} \right\rbrack}} & (11) \\ {{Iy} = {\frac{4}{3}\left\lbrack {{- \frac{{IA} + {IB}}{2}} + {IC}} \right\rbrack}} & (12) \end{matrix}$

The circuit shown in FIG. 32B also has the ability to adjust any of the voltages VS1, VS2, and VS3 independently to any set of voltages within the conversion range of the DC-DC converter. Independent regulation is a function of the duty cycle of the switching waveform at Vg1 and Vg2. Unlike switched capacitor DC-DC converters, buck or boost converters can change the conversion ratio without sacrificing efficiency. This may allow the converter to force each string of cells to its individual MPV and MPP. In this case VS1, VS2, and VS3 may be substantially different if the respective MPVs are different. This ability is useful if the strings of cells are constructed from different materials, different numbers of cells, or have any other characteristic that causes differences in the MPV among strings.

The above exemplary embodiments have been described as applying to PV power sources. However, other power sources within the scope of the invention may include fuel cells, batteries, and thermoelectric energy conversion devices, including thermoelectric diodes.

The foregoing disclosure has been set forth merely to illustrate the invention and is not intended to be limiting. Since modifications of the disclosed embodiments incorporating the spirit and substance of the invention may occur to persons skilled in the art, the invention should be construed to include everything within the scope of the appended claims and equivalents thereof. 

1. An apparatus for equalizing voltages across a plurality of photovoltaic units connected in series, the apparatus comprising: a plurality of energy storage devices, wherein: in a first configuration, each of the energy storage devices is configured to be connected in parallel with one of a first set of the photovoltaic units, and in a second configuration, each of the energy storage devices is configured to be connected in parallel with one of a second set of the photovoltaic units; and a plurality of switches configured to switch between the first configuration and the second configuration, to equalize the voltages across the photovoltaic units, wherein: each of the energy storage devices comprises a circuit having an impedance that is resonant at one or more frequencies, and the circuit comprises an inductor and a capacitor.
 2. The apparatus of claim 1, wherein the inductor and the capacitor are connected in series.
 3. The apparatus of claim 2, wherein the inductor comprises at least one of a discrete inductor on a printed circuit board, interconnect layers of a printed circuit board, interconnect layers of an integrated circuit, bondwires that connect an integrated circuit and a printed circuit board, and interconnect wiring that connects printed circuit boards.
 4. The apparatus of claim 1, wherein the capacitor has a substantial effective series inductance.
 5. The apparatus of claim 1, further comprising a controller configured to output a drive signal to the switches at a switching frequency to control switching of the switches.
 6. The apparatus of claim 5, wherein the switching frequency or a harmonic of the switching frequency equals a resonant frequency of at least one of the energy storage devices.
 7. The apparatus of claim 1, wherein each of the photovoltaic units comprises a photovoltaic cell.
 8. The apparatus of claim 1, wherein each of the photovoltaic units comprises a string of photovoltaic cells that are connected in series.
 9. The apparatus of claim 1, wherein each of the photovoltaic units comprises a module that includes at least one string of photovoltaic cells that are connected in series.
 10. The apparatus of claim 1, further comprising: a plurality of bypass energy storage devices, each of which is configured in parallel with one of the photovoltaic units.
 11. The apparatus of claim 10, wherein the bypass energy storage devices are capacitors.
 12. The apparatus of claim 1, wherein each of the switches comprises a single-pole, double-throw switch that includes at least one MOSFET.
 13. The apparatus of claim 1, further comprising: a power point tracking converter that identifies an aggregate maximum power point for the plurality of photovoltaic units; wherein the voltages across the photovoltaic units are determined based on the aggregate maximum power point.
 14. The apparatus of claim 13, wherein: the power point tracking converter identifies an individual maximum power point for each of the plurality of photovoltaic units; and the voltages across the photovoltaic units are determined based on the aggregate maximum power point and the individual maximum power points.
 15. The apparatus of claim 13, wherein: the power point tracking converter identifies the aggregate maximum power point for the plurality of photovoltaic units and another set of photovoltaic units that is connected in parallel with the plurality of photovoltaic units.
 16. The apparatus of claim 1, further comprising: a connecting circuit configured to provide a connection to an external apparatus, the connecting circuit comprising: a connecting energy storage device; and a connecting switch connected to the connecting energy storage device; wherein: in the first configuration, the connecting energy storage device is configured to be connected in parallel with a photovoltaic unit of the external apparatus via the connecting switch, and in the second configuration, the connecting energy storage device is configured to be connected in parallel with one of the photovoltaic units of the apparatus via the connecting switch.
 17. The apparatus of claim 16, wherein the connecting circuit further comprises: a connecting bypass energy storage device, which is configured to filter the voltage across the photovoltaic unit of the apparatus that is connected in parallel with the connecting energy storage device in the second configuration.
 18. A multi-stage apparatus for equalizing voltages across a plurality of photovoltaic modules connected in series, wherein each of the photovoltaic modules comprises a plurality of photovoltaic units connected in series, the multi-stage apparatus comprising: a plurality of apparatuses, each of which comprises: a plurality of energy storage devices, wherein: in a first configuration, each of the energy storage devices is configured to be connected in parallel with one of a first set of the photovoltaic units within one of the modules, and in a second configuration, each of the energy storage devices is configured to be connected in parallel with one of a second set of the photovoltaic units within the one of the modules; a plurality of switches configured to switch between the first configuration and the second configuration; and a connecting circuit configured to provide a connection between the apparatus and an adjacent apparatus among the plurality of apparatuses, the connecting circuit comprising: a connecting energy storage device; and a connecting switch connected to the connecting energy storage device; wherein: in the first configuration, the connecting energy storage device is configured to be connected in parallel with a photovoltaic unit of the adjacent apparatus via the connecting switch, and in the second configuration, the connecting energy storage device is configured to be connected in parallel with one of the photovoltaic units of the apparatus via the connecting switch; and a plurality of connectors configured to connect each of the apparatuses to an adjacent one of the apparatuses; wherein: switching between the first configuration and the second configuration equalizes the voltages across the photovoltaic units of all of the photovoltaic modules, each of the energy storage devices comprises a circuit having an impedance that is resonant at one or more frequencies, and the circuit comprises an inductor and a capacitor.
 19. A method of equalizing voltages across photovoltaic units connected in series, the method comprising: in a first configuration, storing a first energy from the photovoltaic units in a plurality of energy storage devices, wherein each of the energy storage devices is connected in parallel with one of a first set of the photovoltaic units; in a second configuration, storing a second energy from the photovoltaic units in the plurality of energy storage devices, wherein each of the energy storage devices is connected in parallel with one of a second set of the photovoltaic units; and switching between the first configuration and the second configuration until the voltages across the photovoltaic units are substantially equal, wherein: each of the energy storage devices comprises a circuit having an impedance that is resonant at one or more frequencies, and the circuit comprises an inductor and a capacitor.
 20. The method of claim 19, wherein a frequency of the switching between the first configuration and the second configuration equals a resonant frequency of the energy storage devices.
 21. The method of claim 19, further comprising: after the switching between the first configuration and the second configuration, sending an energy equal to a difference between the first energy and the second energy to a plurality of bypass energy storage devices.
 22. The method of claim 19, further comprising: identifying an aggregate maximum power point for the plurality of photovoltaic units; and determining the voltages across the photovoltaic units based on the aggregate maximum power point.
 23. The method of claim 22, further comprising: identifying an individual maximum power point for each of the plurality of photovoltaic units; wherein the voltages across the photovoltaic units are determined based on the aggregate maximum power point and the individual maximum power points.
 24. The method of claim 22, wherein the aggregate maximum power point is identified for the plurality of photovoltaic units and another set of photovoltaic units that is connected in parallel with the plurality of photovoltaic units.
 25. An apparatus for equalizing voltages across a plurality of photovoltaic units connected in series, the apparatus comprising: a plurality of energy storage devices, wherein: in a first configuration, a first energy storage device is configured to be connected in parallel with a first photovoltaic unit, and a second energy storage device is configured to be connected in parallel with a second photovoltaic unit; and in a second configuration, the first energy storage device is configured to be connected in parallel with the second photovoltaic unit, and the second energy storage device is configured to be connected in parallel with a third photovoltaic unit; and a plurality of switches configured to switch between the first configuration and the second configuration, to equalize the voltages across the photovoltaic units, wherein: each of the energy storage devices comprises a circuit having an impedance that is resonant at one or more frequencies, and the circuit comprises an inductor and a capacitor.
 26. An apparatus for equalizing voltages across a plurality of photovoltaic units connected in series, the apparatus comprising: an energy storage device, wherein: in a first configuration, the energy storage device is configured to be connected in parallel with a first one of the photovoltaic units, and in a second configuration, the energy storage device is configured to be connected in parallel with a second one of the photovoltaic units; and a switch configured to switch between the first configuration and the second configuration, to equalize the voltages across the photovoltaic units, wherein: the energy storage device comprises a circuit having an impedance that is resonant at one or more frequencies, and the circuit comprises an inductor and a capacitor. 